Some Chemistry Indices of Clique-Inserted Graph of a Strongly Regular Graph

In this paper, we give the relation between the spectrum of strongly regular graph and its clique-inserted graph. The Laplacian spectrum and the signless Laplacian spectrum of clique-inserted graph of strongly regular graph are calculated. We also give formulae expressing the energy, Kirchoff index, and the number of spanning trees of clique-inserted graph of a strongly regular graph. And, clique-inserted graph of the triangular graph T t , which is a strongly regular graph, is enumerated.

AUTOMATIC DATA CLASSIFICATION BASED ON THE TRIANGULAR GRAPH FOR THEMATIC MAPS

A triangular point graph helps in the process of data classification for a thematic map. A triangular graph can be used for a situation that is described by three variables. The total sum of variables is 100%. The proportion of three variables is plotted in an equilateral triangular graph where each side represents a coordinate for one variable. A triangular graph displays the proportions of the three variables. The position of the point indicates the type (class) of the situation in the triangular graph. The typology of the situation can be subsequently expressed in the map. We have created a “Triangular Graph” program which represents a helpful automatic tool for ArcGIS software. This new program classifies input data based on a triangular graph. It is realized by two python scripts located in a custom toolbox as two programs. The first program calculates X and Y coordinates in an equilateral triangular graph. The second program compares plotted points and suggested zones of a division produced by the first program. Finally, a new attribute is added to the source data. The user can create a new thematic map, based on that attribute in order to express the typology of the given situation. The programming language Python and essential module ArcPy have been used for solving these tasks. To test the created programs several maps were made, based on the classification often used in demography. For example, the new program helped to create a sample map of age categories in districts of the Czech Republic. The program is available to download from the Esri web pages and web pages of the Department of Geoinformatics, Palacký University Olomouc.

On the Shannon Capacity of Triangular Graphs

The Shannon capacity of a graph $G$ is $c(G)=\sup_{d\geq 1}(\alpha(G^d))^{\frac{1}{d}},$ where $\alpha(G)$ is the independence number of $G$. The Shannon capacity of the Kneser graph $\mathrm{KG}_{n,r}$ was determined by Lovász in 1979, but little is known about the Shannon capacity of the complement of that graph when $r$ does not divide $n$. The complement of the Kneser graph, $\overline{\mathrm{KG}}_{n,2}$, is also called the triangular graph $T_n$. The graph $T_n$ has the $n$-cycle $C_n$ as an induced subgraph, whereby $c(T_n) \geq c(C_n)$, and these two families of graphs are closely related in the current context as both can be considered via geometric packings of the discrete $d$-dimensional torus of width $n$ using two types of $d$-dimensional cubes of width $2$. Bounds on $c(T_n)$ obtained in this work include $c(T_7) \geq \sqrt[3]{35} \approx 3.271$, $c(T_{13}) \geq \sqrt[3]{248} \approx 6.283$, $c(T_{15}) \geq \sqrt[4]{2802} \approx 7.276$, and $c(T_{21}) \geq \sqrt[4]{11441} \approx 10.342$.